Ternary Relational Semantics for the Variants of BN4 and E4 which Contain Routley and Meyer's Logic B

Six hopefully interesting variants of the logics BN4 and E4 – which can be considered as the 4-valued logics of the relevant conditional and (relevant) entailment, respectively – were previously developed in the literature. All these systems are related to the family of relevant logics and contain Routley and Meyer's basic logic B, which is well-known to be specifically associated with the ternary relational semantics. The aim of this paper is to develop reduced general Routley-Meyer semantics for them. Strong soundness and completeness theorems are proved for each one of the logics.